theoretical analysis of a dipole-fed horn antennadiscontinuities to be included in the horn geometry...

First International Symposium on Space Terahertz Technology Page 187

THEORETICAL ANALYSIS OF A DIPOLE-FED HORN

ANTENNA

G.V. Eleftheriades, Walid Ali-Ahmad, L.P.B. Katehi, G.M Rebeiz

Center for Space Terahertz Technology,

University of Michigan, Ann Arbor, MI

March 1990

Abstract

This antenna consists of a strip dipole printed on a membrane and suspended in an etched

pyramidal silicon cavity. The entire fabrication is monolithic. For the theoretical character-

ization, the horn is approximated by a stepped structure of multiple rectangular waveguicle

sections. The fields in each section are given by a linear combinatión of waveguide modes,

and the fields in air are given by a continuous plane wave spectrum. To evaluate the Green's

function for this problem, an infinitesimally small electric dipole is considered in one of the

waveguide sections. Then, the fields inside the silicon cavity are evaluated, and the far-field

patterns are found from the Fourier transforms of the electric field on the aperture of the

horn.

First International Symposium on Space Terahertz Technology

1. INTRODUCTION

In millimeter wave imaging systems the use of a single detector with electronic or me-

chanical scanning is inadequate. The events may be too fast, or the required integration

time too long. The way to eliminate this limitation is to image all points simultaneously.

In 1987, G.M. Rebeiz fabricated a monolithic millimeter-wave imaging array consisting of a

large number of silicon horns with detectors placed at the focal plane of the imaging system.

For the design of this array, the radiating elements were characterized experimentally and

mutual coupling was neglected. As a result, the efficiency of the array was approximately

45 percent. To improve the performance of the array, the elements have to be modeled

accurately. This paper presents the development of a rigorous method for the theoretical

characterization of the isolated elements.

The procedure for rigorously generating the Green's function for a dipole-fed metallic

rectangular horn radiating in free-space is being described. The method allows for step

discontinuities to be included in the horn geometry and is not limited by large flaring

angles or short axial lengths.

2. THEORY

The procedure which has been employed in the formulation of the problem consists of

five steps:

1. The geometry of the horn is approximated by a cascade of rectangular waveguide

steps. One of the waveguide steps contains an infinitesimal dipole excitation.

2. The field on the apex and aperture of the horn is transfered, via appropriate trans-

mission matrices, on the excitation (source) plane.

3. The aperture field is matched to free space using plane-wave expansion.

4. The transfered fields of step two are matched over the excitation plane.

5. Finally steps (3) and (4) combined together result in a linear system of equations.

The solution of this system provides the required Green's function in discrete form

inside the horn and in continuous form in the half space.


2.1 Approximation of the horn geometry

The horn is approximated by a multistepped discontinuity consisting of N sections as

shown in Figure 1.

The horn aperture lies on an infinite-metallic screen. Furthermore, it is assumed that

there is no reflection from the apex of the horn, since any reflection will be at cut-off.

2.2 Transfer of the apex and Apperture Field on the source plane

Each step discontinuity is characterized by a transmission matrix [Tb]

B(2)

D(2)

A(2)

61(2)

= [Tbi (1)

where the elements of the column matrices denote the unknown coefficients of the field next

to the discontinuity. The field in each waveguide is represented in a modal expansion as

shown below :

004i) [-e TE(x, y) { A (;7:ione_,I,ncon

, B(i) „0,)771) z}rnn

m,,n

y) Cti)ne—'41)nz e+')Vnz}] (2)

- x Ht(i) • E [e.2)7,1TE (x, 074T E, A (i) pie p-y2.)r z

m ,n

+ (x, — D (i) e"'Uz } ".mm mm mnmm

where i takes the values 1 or 2.

In the above representation both TE and TM modes are included to support the analysis

of an arbitrary planar excitation current. A cascade of such transmission matrices, along

with associated phase delay transmission matrices is used to transfer the apex and aperture

field on the excitation plane :

(3)

- apex apperture

= [TB]0

0

-BII

DLL

An

CH

(4)= [TA]

-B(N)

D(N)

A(N)

C(N)

F11 F12 F13 F12

F21 F22 F21 F24

0

0(7)

0

0

°I° (k2 V' • • (k k )eln(k ky)dkdkx y1_00 z xi3 x,rPqlijmn (8)

Page 190 First International Symposium on Space Terahertz Technology

Referring to fig.1, the source interface divides the source section #K into two subsections.

one on the left, having length 1, and one on the right, having length 1 H . Accordingly, in

equations (4) above, the modal coefficients of the left subsection are denoted with the index

(I) and those of the right subsection with the index (II).

2.3 Matching to Half space

This task matches the discrete field spectrum of the aperture section of the horn to the

continuous field spectrum of free space, over the horn aperture. The field in free-space is

expressed as a plane wave superposition:

1 00 oo yyE(x , y, z)f_

, ky)e—ik ik e- ikzz dkrdky2r oo

f

with the transverse component of (lc, ky ) given by:

(5)

gt(k ky) =1

27r I /apertureEt (x , y , 0+) e ik.rx ikyye dxdy

Field continuity on the aperture of the horn leads to the following matrix equation

The submatrices Fii contain reaction integrals involving the discrete modes of the aper-

ture field. There are three kind of reaction integrals as shown below

B(H) 0

D(II)

A(H) ern(xi, YI)r-,(11)

eInimn(xl

(15)[A] = [A]


°13 r (k2 _ k!) e..7) (k kx y),e y*o x, y ) yk dk, dk (9)2ijmn00 — 00 Z

/Pqkxky—[e • -(ks ky)e" (ks, ky)

eP• •(

kx,

ky)

e" (k k y Adk dk &Of))3ijmn 00 _00 kz yzj xmn ri ymn

where p,q take the values 1 (—÷ TE) or 2 (---> TM).

In order to facilitate the numerical evaluation of the integrals it is advantageous to

transform them in the space-domain. Their transformation from the spectral-domain to

the spatial-domain is accomplished via the application of Parseval's theorem leading to

ITN TYN?. — (k2Jrv —)l mn ay2Lilt T

N i—

YN P

[ePsiii (—x) eximm (x)j{ePx2ii (—y) el2mn(y)]dxdy

pg . 1 fxN p N ie —ikp ,92

213mn — (k2xN i— YN P

ax2 Ylij( —X ) eylmn(X)]

[e 2 ( — y) ev2mn(Y)NXdy (12)

rpq 1 fxN ie—ikp[(ePi..(—x) eximn(x))1 3ijmn =

`i " X N P axay Y

( ePy2ij( — Y) eqs2ii(Y)) ( eP X eYlmn(X))(ePT2ij(—Y)xlij ( —) eyq2mn(y)]dxdy (13)

where p = Vx2 + y2.

2.4 Matching on the source interface

The point excitation is assumed to be a y-directed surface current given by:

K(x,y) = 6(x — /)6(y — (14)

Upon matching over the source interface we derive the matrix equation

B(1)

Do-)

B- (N)

D(N)

A(N)

C(N)

(17)= [TA]= [TB]0

0appertureapex

• Matching of the aperture field :

[A] = [A]

0

ern,E ri (x' , y1)

eLmn (x' ,

(19)


where matrix [A] above involves the mode admittances of the source section as shown

below :

0_yTE,K 0 y TE,K

(16)

_yTM,K Q y TM,K

2.5 Derivation of the final system of equations

The equations that have been generated so far are summarized below

• Apex-aperture field transfered on the source plane through transmission matrices:

B (N) 0

D(N)=-_

A(N ) 0

C(N) 0

▪ Matching of the fields on the source interface

[1] (18)

[-TEr \ -TM/ \ -TE/ \ -TMemnAx, y) x,y) emnAx,y) emn [TB] [S Ili 1]

0

eTmEri (x', yi)

ernT

14. (xl yi)

(21)


The above equations are merged together to form a single matrix equation involving the

modal coefficients of the apex and aperture sections

DO-)

{ill[T B y — [A][11

A l B (N ) el

0 [1] D(N) €2

.4 (N) 0

C(N) 0

Note that the block-submatrix ([A][m]y consists only of the half first columns of [A][TE31.

This is a consequence of the assumption of no reflections from the apex of the horn. Having

found the modal coefficients of the apex and aperture sections, the Green's function inside

the horn is represented as a discrete Fourier series whereas outside the horn as a Fourier

integral. Specifically the transverse component of the Green's function for a y-directed

current inside the horn can be represented in a quadratic form as shown below

G t (x , y ,

(20)

In the above representation [SIM] is an appropriate submatrix of the inverse of the

system matrix defined in (19).

3. NUMERICAL RESULTS

The aperture field and the far-field for a specific horn geometry has been evaluated. For

this purpose the aperture coefficients are calculated from the solution of matrix equation

(20). For a vertical point source only modes of the form (m,n) [ m=1,3,5, ... and n=0,2,4,

...] are excited. Numerical convergence for the far-field was achieved when TE and TM

modes up to the order (7,7) were included. The magnitude of the co-polarized aperture field


shown in fig.4 is recognized to be a perturbed dominant mode field distribution. Furter-

more,the utilization of both TE and TM modes in the theory enabled the calculation of the

cross-polarized aperture field which has a level of -16dB for the case under consideration.

The phase of the aperture field is depicted in fig.5. Of interest is the cross-polarization phase

which divides the aperture ,along its geometrical axes of symmetry, into four quadrants.

Each quadrant is out of phase with its adjacent quadrants.

4. CONCLUSION

This paper presented a rigorous method for the characterization of a monolithic horn

antenna excited by an infinitesimal small dipole printed on a membrane. Using this method,

the fields on the aperture of the silicon horn were evaluated and transformed to the Fourier

domain in order to derive the far-field patterns. The developed method may easily be

extended to compute input impedance and resonant properties of these radiating structures

for feeding dipoles of finite size.

5. AKNOWLEDGMENTS

The work was sponsored by the Center for Space Terahertz Technology, the University

of Michigan, Ann Arbor, MI.


REFERENCES

1. Thomas Wriedt, Karl-Heinz Wolff, Frint Arndt and Ulrich Tucholke "Rigorous Hy-

brid Field Theoretic Design of Stepped Rectangular Waveguide Mode Converters In-

cluding the Horn Transitions into Half-Space," IEEE Transactions on Antennas and

Propagation, Vol. 37, No. 6, June 1989.

2. Amir I. Zaghloul and Robert H. MacPhie, "Cross-Correlation Formulation of the

Complex Power from Planar Apertures," Radio Science, Vol. 10, No. 6, pp. 619-624,

June 1975.

3. Robert H. McPhie and Amir I. Zaghloul, "Radiation from a Rectangular Waveguide

with Infinite Flange-Exact Solution by the Correlation Matrix Method," IEEE Trans-

actions of Antennas and Propagation, Vol. AP-28, No. 4, July 1980.

4. Jose A. Encinar and Jesus Rebollar, "A Hybrid Technique for Analyzing Corrugated

and Non-Corrugated Rectangular Horns," IEEE Transactions on Antennas and Prop-

agation, Vol. AP-34, No. 8, August 1986.

(1)

C(1)

„so #1

D(1) b1

440-- B(2)a2 D(2)

Y2

%xh ...-0■■• A(2)

1,ir c(2)

,g,0,0

#2


Fig.1 A waveguide step discontinuity.

First-section#1

(Apex)

excitationinterface

Last-Section(#N)(Aperture)


Infinitemetallicscreen

A-- --Øsr :I

••

A z •

•• •

il 1 1• ••

•I -÷:

•• A4—

I :

# N•

, • •, a

1 TT1

b N

1

I".. 1

0: 44-1". 11 Jr....4o.

I II

0

b 1•1 0 •-4----, •• • •

••#1 • ,• •

I•• •

• •

I••

•• •

• 4•11---'•

LiI

#K •••

•

• ••••

•• i

•

••

•

• 1 •• •

I •••

Half-space

Fig.2 Approximation of the horn geometry by cascaded step discontinuities.

0.00

-20.

-25.-90. -75. -60. -45. -30. 45. 0.00 15. 30. 45. 60. 75. 90.

Page 198 First International Symposium on Spaee Terahertz Technology

FAR-FIELD

THETA (DEG)

Aperture Dimensions -

1.45 A X 1.45 A

Flaring Angles

70°

Diplole Polarization

Vertical (along y-axis,,

Dipole Position

0.38A from apex

Fig.3 Far-field.


MAGNITUDE OF E-FIELD ALONG APERTURE DIAGONAL1.00

0.90

0.80

0.70

060

0.50

.

0.40

0.30

0.20

0.10

0.000.000 0.210 0.420 0.630 0.840 1.050 1.260 1.470 1.680 1.890 2.100

distance along aperture diagonal

Fig.4 Magnitude of aperture field. (a) The co-polarazized field. (b) The

cross-polarized field. (c) Comparison of the co-polar and cross-polar magni-

tude along the aperture diagonal.


phaso(Ey)

PHASE OF E -FIELD ALONG APERTURE Y-AXIS360. - - -330.

300.

210.

240.

180.

150.

120. "•

Ey

7

7

270.

90. Ex6o. -30.

0.000 0.145 0.290 0.435 0.580 0.725 0.870 1.015 1.160 1.305 1.450

distance along y-axis (wavelengths)

Fig.5 Phase of aperture field. (a) The co-polarized field. (b) The cross-

polarized field. (c) Comparison of the co-polar and cross-polar phase along

the aperture diagonal.

theoretical analysis of a dipole-fed horn antennadiscontinuities to be included in the horn geometry...

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